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Properties

Label 350.5.k.a
Level $350$
Weight $5$
Character orbit 350.k
Analytic conductor $36.179$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [350,5,Mod(101,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.101");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 350.k (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(36.1794870793\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - 2 \beta_{3} + 3 \beta_{2} + \cdots + 6) q^{3}+ \cdots + (42 \beta_{2} + 36 \beta_1 + 42) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - 2 \beta_{3} + 3 \beta_{2} + \cdots + 6) q^{3}+ \cdots + (1224 \beta_{3} - 2862) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 18 q^{3} - 16 q^{4} + 28 q^{7} + 84 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 18 q^{3} - 16 q^{4} + 28 q^{7} + 84 q^{9} - 54 q^{11} - 144 q^{12} - 128 q^{16} - 918 q^{17} - 576 q^{18} + 30 q^{19} - 378 q^{21} - 192 q^{22} + 486 q^{23} - 768 q^{24} - 1728 q^{26} - 1456 q^{28} + 3240 q^{29} - 546 q^{31} - 1062 q^{33} - 1344 q^{36} + 446 q^{37} + 4320 q^{38} - 3312 q^{39} - 5376 q^{42} - 2344 q^{43} - 432 q^{44} + 3840 q^{46} - 702 q^{47} - 9212 q^{49} + 318 q^{51} + 384 q^{52} - 2754 q^{53} - 1440 q^{54} + 17460 q^{57} - 384 q^{58} + 12366 q^{59} + 7686 q^{61} - 6468 q^{63} + 2048 q^{64} - 3456 q^{66} + 5062 q^{67} + 7344 q^{68} + 18792 q^{71} - 4608 q^{72} + 17274 q^{73} - 4320 q^{74} - 4914 q^{77} - 8832 q^{78} + 794 q^{79} - 4338 q^{81} - 9984 q^{82} - 5040 q^{84} - 10368 q^{86} + 12276 q^{87} + 768 q^{88} - 12474 q^{89} + 2688 q^{91} - 7776 q^{92} - 18918 q^{93} + 15168 q^{94} + 6144 q^{96} - 11448 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1 + \beta_{2}\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
−0.707107 − 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 − 1.22474i
−1.41421 − 2.44949i −3.98528 − 2.30090i −4.00000 + 6.92820i 0 13.0159i 7.00000 + 48.4974i 22.6274 −29.9117 − 51.8086i 0
101.2 1.41421 + 2.44949i 12.9853 + 7.49706i −4.00000 + 6.92820i 0 42.4098i 7.00000 + 48.4974i −22.6274 71.9117 + 124.555i 0
201.1 −1.41421 + 2.44949i −3.98528 + 2.30090i −4.00000 − 6.92820i 0 − 13.0159i 7.00000 − 48.4974i 22.6274 −29.9117 + 51.8086i 0
201.2 1.41421 − 2.44949i 12.9853 − 7.49706i −4.00000 − 6.92820i 0 − 42.4098i 7.00000 − 48.4974i −22.6274 71.9117 − 124.555i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.5.k.a 4
5.b even 2 1 14.5.d.a ✓ 4
5.c odd 4 2 350.5.i.a 8
7.d odd 6 1 inner 350.5.k.a 4
15.d odd 2 1 126.5.n.a 4
20.d odd 2 1 112.5.s.b 4
35.c odd 2 1 98.5.d.a 4
35.i odd 6 1 14.5.d.a ✓ 4
35.i odd 6 1 98.5.b.b 4
35.j even 6 1 98.5.b.b 4
35.j even 6 1 98.5.d.a 4
35.k even 12 2 350.5.i.a 8
105.o odd 6 1 882.5.c.b 4
105.p even 6 1 126.5.n.a 4
105.p even 6 1 882.5.c.b 4
140.p odd 6 1 784.5.c.b 4
140.s even 6 1 112.5.s.b 4
140.s even 6 1 784.5.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.5.d.a ✓ 4 5.b even 2 1
14.5.d.a ✓ 4 35.i odd 6 1
98.5.b.b 4 35.i odd 6 1
98.5.b.b 4 35.j even 6 1
98.5.d.a 4 35.c odd 2 1
98.5.d.a 4 35.j even 6 1
112.5.s.b 4 20.d odd 2 1
112.5.s.b 4 140.s even 6 1
126.5.n.a 4 15.d odd 2 1
126.5.n.a 4 105.p even 6 1
350.5.i.a 8 5.c odd 4 2
350.5.i.a 8 35.k even 12 2
350.5.k.a 4 1.a even 1 1 trivial
350.5.k.a 4 7.d odd 6 1 inner
784.5.c.b 4 140.p odd 6 1
784.5.c.b 4 140.s even 6 1
882.5.c.b 4 105.o odd 6 1
882.5.c.b 4 105.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 18T_{3}^{3} + 39T_{3}^{2} + 1242T_{3} + 4761 \) acting on \(S_{5}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$3$ \( T^{4} - 18 T^{3} + \cdots + 4761 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 14 T + 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 54 T^{3} + \cdots + 194481 \) Copy content Toggle raw display
$13$ \( T^{4} + 62592 T^{2} + 955551744 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 2084013801 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 37762205625 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 161403866001 \) Copy content Toggle raw display
$29$ \( (T^{2} - 1620 T + 651492)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 566643101049 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 284591307841 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 1046087110656 \) Copy content Toggle raw display
$43$ \( (T^{2} + 1172 T - 3015836)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 5548271897529 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 8389590597441 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 149611524833241 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 66399241936329 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 256392053989009 \) Copy content Toggle raw display
$71$ \( (T^{2} - 9396 T + 20407716)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 29440640401929 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 54769812839569 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 314640617324544 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 28434692391561 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
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